How to show that $R_{\mathfrak{m}}$ is $R$?

Question

Let $R$ be a discrete valuation ring and $\mathfrak{m}$ its unique non-zero maximal ideal. How to show that $R_{\mathfrak{m}}$ is $R$ using definition of a discrete valuation ring? I know that elements of $R_{\mathfrak{m}}$ is of the form $(r, s)$, $s \in R-\mathfrak{m}$. Thank you very much.

Answer 1

This is a property of all local rings $(R,\mathfrak m)$. To see why: recall that $\mathfrak m$ is the set of all noninvertible elements of $R$; thus, any element that would be a denominator in $R_\mathfrak m$ is already invertible.